Problem: Solve for $x$ : $ 2|x - 5| + 3 = 3|x - 5| + 4 $
Solution: Subtract $ {2|x - 5|} $ from both sides: $ \begin{eqnarray} 2|x - 5| + 3 &=& 3|x - 5| + 4 \\ \\ {- 2|x - 5|} && {- 2|x - 5|} \\ \\ 3 &=& 1|x - 5| + 4 \end{eqnarray} $ Subtract $4$ from both sides: $ \begin{eqnarray} 3 &=& 1|x - 5| + 4 \\ \\ {- 4} && {- 4} \\ \\ -1 &=& 1|x - 5| \end{eqnarray} $ Simplify: $ -1 = |x - 5| $ The absolute value cannot be negative. Therefore, there is no solution.